A strong law of large numbers for super-stable processes

作者：Kouritzin Michael A; Ren Yan Xia^*

来源：Stochastic Processes and Their Applications, 2014, 124(1): 505-521.

DOI：10.1016/j.spa.2013.08.009

• Super-stable process
• Super-Brownian motion
• Strong law of large numbers
• Fourier transform
• Vague convergence
• Probability measures

Let be Lebesgue measure and X = (X-1, t >= 0; P mu) be a supercritical, super-stable process corresponding to the operator 7 (-Delta)(alpha/2) u + beta u eta u(2) on Rd with constants beta, eta > 0 and a alpha is an element of (0, 2]. Put W-1 (theta) = e((theta)) r (-1 theta) X-t(e(-i theta)), which for each small theta is an a.s. convergent complex-valued martingale with limit a W(theta) say. We establish for any starting finite measure mu satisfying f(R)(d) vertical bar x vertical bar mu(dx) < infinity that t(d)/alpha W(0) c alpha W (0)l P mu-a.s. in a topology, termed the shallow topology, strictly stronger than the vague ePt topology yet weaker than the weak topology, where c(alpha) > 0 is a known constant. This result can be thought of as an extension to a class of superprocesses of Watanabe's strong law of large numbers for branching Markov processes.

• 出版日期2014-1
• 单位北京大学